Non-synchronous belt driven camshaft phase shift device

ABSTRACT

A non-synchronous camshaft phasing device  46  for use with an internal combustion engine E. The internal combustion engine E includes an engine control unit ECU, a camshaft  42  and a crankshaft  12.  The non-synchronous phasing device  46  is located between the crankshaft  12  and the camshaft  42  for controlling a phase shift angle between the camshaft  42  and the crankshaft  12.  The phasing device  46  comprises an input shaft  36  coupled to the crankshaft  12  via a non-synchronous belt  40.  The phasing device  46  also comprises an output shaft  42  coupled to the camshaft  44;  a planetary gear train  48  co-axially aligned around and coupled with the input shaft  36  and the output shaft  42;  and an motor  50  coupled to the planetary gear train  48  by a carrier  56.  A controller operatively connects to the engine control unit ECU, wherein the controller is configured to receive engine operating signals generated by the engine control unit ECU and to receive signals from position sensors  51  coupled to the input shaft  36  and to the output shaft  42.  In response to the signals, the controller generates and sends a torque command signal to the motor  50  to command the motor  50  to control the planetary gear train  48  through the carrier 56 to adjust the phase shift angle between the camshaft and the crankshaft  12.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is related to, and claims priority from, U.S. Provisional Patent Application No. 60/978,568 filed on Oct. 9, 2007, herein incorporated by reference.

TECHNICAL FIELD

Camshaft phase shifting devices are used in internal combustion engines to vary valve timing to improve fuel consumption and to improve exhaust gas quality. It is possible with current camshaft shifters to time the operation of the valves for maximum comfort and/or for maximum torque and the highest performance. Camshaft phase shifting devices used today are driven by a crankshaft though a synchronous belt or chain drive. The use of positive/synchronous engagement drive systems (i.e. toothed belt drives and chain drives) is due primarily to the stringent timing requirement between the crankshaft and the camshaft. The cost, however, associated with positive engagement drive systems is higher than that of the non-positive engagement drive systems such as flat belt or V-belt drive systems, known as non-synchronous belts.

It is desirable to have a camshaft phasing device that is suitable for being driven by a simple non-positive/non-synchronous belt drive for packaging and cost savings, and yet is adjustable to achieve and maintain desired valve timing, while being electronically controlled for simplicity and high precision.

BRIEF SUMMARY OF THE DISCLOSURE

Briefly stated, the present disclosure relates to a camshaft phase device for an internal combustion engine, and in particular, relates to a non-synchronous, belt driven camshaft phase device.

The belt driven camshaft phase device comprises a non-synchronous belt and an epiclyclic gear train operatively connected to an input shaft and an output shaft. The input shaft is connected to the crankshaft via the non-synchronous belt and the output shaft is connected to a camshaft. The camshaft phase device further includes sensors and a controller, through which the positions of the input and output shafts and the positions of the camshaft and crankshaft are detected and tracked. Should the desired relationship in positions between the crankshaft and camshaft become unsynchronized as determined by an error signal exceeding a tolerance band, correction or compensation is applied to the output shaft through the gear train. The camshaft phase device of the present disclosure includes an adequate slew rate to achieve real-time compensation for mismatches in relative angular positions between the camshaft and crankshaft resulting from the operation of the non-synchronous belt drive system.

The foregoing features, and advantages set forth in the present disclosure as well as presently preferred embodiments will become more apparent from the reading of the following description in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

In the accompanying drawings which form part of the specification:

FIG. 1 is a schematic view of a non-synchronous, belt-driven drive system and schematically illustrates internal components of an internal combustion engine, associated pulleys thereof and a non-synchronous belt;

FIG. 2 is a schematic view of a camshaft phase shift device constructed in accordance with and embodying the present disclosure;

FIG. 3 illustrates a cross sectional view of the input shaft, an output shaft and the phase shift device;

FIG. 4 illustrates an exploded view of a phase shift device constructed in accordance with and embodying the present disclosure;

FIG. 5 illustrates another exploded view of components of the phase shift device of FIG. 4;

FIG. 6 illustrates a cross sectional view of a phase shift device constructed in accordance with and embodying the present disclosure; and

FIG. 7 is a schematic view of a torque based control structure of the camshaft phase shift device that controls the desired angular position of the output shaft of with respect to the input shaft.

Corresponding reference numerals indicate corresponding parts throughout the several figures of the drawings. It is to be understood that the drawings are for illustrating the concepts set forth in the present disclosure and are not to scale.

DETAILED DESCRIPTION

The following detailed description illustrates the invention by way of example and not by way of limitation. The description enables one skilled in the art to make and use the present disclosure, and describes several embodiments, adaptations, variations, alternatives, and uses of the present disclosure, including what is presently believed to be the best mode of carrying out the present disclosure. Referring to the drawings, a drive system for an internal combustion engine E is schematically shown as 10 (FIG. 1). The drive system comprises a crankshaft 12 and crankshaft pulley 14; an air-conditioning compressor 16 and compressor pulley 18; a power steering pump 20 and pump pulley 22; a water pump 24 and pump pulley 26; an alternator 28 and alternator pulley 30; tensioner 32 and tension pulley 34; input shafts 36 and associated pulleys 38 and a non-synchronous belt 40. The non-synchronous belt 40 operatively connects associated pulleys 14, 18, 22, 26, 30, 34 and 38 wherein the crankshaft 12, via its pulley 14, drives the non-synchronous belt 40.

Turning to FIGS. 2-6, the input shaft 36 couples with the input pulley 38 at an end of the input shaft 36. An output shaft 42 couples with a camshaft 44 at an end of the output shaft 42. Further, an electro-mechanic phase shift device of the present disclosure, generally shown as 46, is shown located at the end of a camshaft 44 of the internal combustion engine E. The phase shift device 46 comprises an epicyclic gear train generally shown as 48; a motor generally shown as 50; sensors 51 and associated target wheels 47, 49 in operative connection with the input shaft 36 and the output shaft 42 and an engine control unit ECU.

The epicyclic gear train 48 co-axially aligns around the input shaft 36 and the output shaft 42. The epicyclic gear train 48 comprises a first branch in the form of an input sun gear 52, a second branch in the form of an output sun gear 54, and a control branch in the form a carrier 56. The gear train 48 also comprises a first planet gear 58 and a second planet gear 60. As known in the art, the first planet gear 58 may comprise a set of first planet gears and the second planet gear 60 may comprise a set of second planet gears. Optimally, the sets of first planet gears and second planet gears 60 are equally spaced within the carrier 56.

The input sun gear 52 meshes with the first set of planet gears 58, and the output sun gear 54 meshes with the second set of planet gears 60. Each planet gear 58 in the first planet gear set couples to, and thus rotates as a unit with, a corresponding planet gear 60 in the second planet gear set. Planet gears 58, 60 together form a planetary gear pair to rotate about a common axis at the same angular velocity. The planetary gear pairs are supported by a set of planet shafts 62 (FIG. 2), through bearings 64. The carrier 56 is supported in a housing 66 though bearings 68.

In an embodiment (FIGS. 4-6), planet gears 58, 60 are substantially identically formed and are integrated as a single gear 70. The single gear 70 has a first gear end 72 and a second gear end 74 correlating to planet gears 58, 60, respectively. FIG. 6 illustrates a cross sectional view of a set of single gears 70 positioned 180 degrees apart.

The input shaft 36 connects to input pulley 38 at one end and to the input sun gear 52 at the other end. The input shaft 36 is supported in the housing 66 though bearings 64. The output shaft 42 connects to the output sun gear 54 at one end and couples to camshaft 44 at the other end. The output shaft 42 is supported in the housing 66 through bearings 64. As known in the art, the first and second sun gears 52, 54 may be integrally formed, respectively, from the input shaft 36 and output shaft 42. As shown, the motor 50 includes a rotor 76 and a stator 78. The rotor 76 fits over the carrier 56 to establish a firm mechanical connection, so that the carrier 56 rotates with the rotor 76 as a unit. As shown, the stator 78 mounts to the housing 66.

To improve supporting stiffness, the input shaft 36 and output shaft 42 may extend beyond the input sun gear 52 and the output sun gear 54 with one piloted on the other through bearing 80 (FIG. 2). Input shaft 36 is allowed to rotate with respect to the output shaft 42 when phase shift between the two shafts 36, 42 is desirable. Optimally, to prevent excessive angular displacement between the two shafts 36, 42 an angular position limiting device generally shown as 82 (FIGS. 1, 4-6) is employed to provide mechanical stops in both rotating directions.

The limiting device 82 rotatably couples the input sun gear 52 with the output sun gear 54. Referring to FIGS. 4-6, the limiting device 82, in an embodiment, comprises a slot 84 positioned on a face 86 of the input sun gear 52 and comprises an extension 88 protruding from another face 90 of the output sun gear 54 such that the extension 88 slidably engages with the slot 84. In an embodiment, the extension 88 comprises pins protruding from the output sun gear 54. During rotation of the shafts 36, 42, the extensions 56 slidably reciprocate within the opposing slot 84 such that the slots 84 limit travel movement of the extensions 56 to prevent excessive angular displacement between the shafts 36, 42.

During operation, the crankshaft 12 drives the input shaft 36 via the serpentine belt 40 through crankshaft pulley 14 and input pulley 38. The input shaft 36, in turn, drives the output shaft 42 through the gear train 48. Sensors 51 monitor the angular velocities and positions of the input shaft 36 and output shaft 42 via target wheels 47, 49. The sensors 51 then communicate the shaft information to the engine control unit ECU.

In an embodiment, the effective creep rate, defined as a percentage pitch line velocity loss with respect to pitch line velocity of the crankshaft pulley 14, is denoted below as “γ”. The ratio of pitch diameter of the input shaft pulley 38 to the pitch diameter of the crankshaft pulley 14 is denoted below as “ψ”. The ratio “φ” of the angular velocity of the crankshaft 12 to the angular velocity of the input shaft 36 is characterized as

$\begin{matrix} {\phi = {\frac{\psi}{1 - \gamma} \cdot}} & (1) \end{matrix}$

If the nominal effective creep rate is γ=γ₀, it is optimal to choose the pulley size for the crankshaft 12 and the input shaft 36 such that the resulting angular velocity ratio φ according to equation (1) is substantially close to 2. In other words, the pulley diametric ratio of the input shaft 36 to the crankshaft 12 is set as

ψ₀=2(1−γ₀).   (2)

To ensure the synchronization between the crankshaft 12 and the camshaft 44, the angular speed of the carrier 56 is set in accordance with the angular speed of the input shaft 36 or the output shaft 42 to closely maintain the following relationship

$\begin{matrix} {\frac{\omega_{C}}{\omega_{S\; 1}} = {{\frac{2 - {2i_{b}}}{\phi - {2i_{b}}}\mspace{14mu} {or}\mspace{14mu} \frac{\omega_{C}}{\omega_{S\; 2}}} = \frac{\phi - {2i_{b}}}{\phi - {\phi \cdot i_{b}}}}} & (3) \end{matrix}$

where

-   -   ω_(C)=angular speed of the carrier 56;     -   ω_(S1)=angular speed of the input shaft 36;     -   ω_(S2)=angular speed of the output shaft 42;

i_(b)=base gear ratio of the differential gear train 48, defined as

$\begin{matrix} {i_{b} = \frac{N_{S\; 1} \cdot N_{P\; 2}}{N_{S\; 2} \cdot N_{P\; 1}}} & (4) \end{matrix}$

where

N_(S1), N_(S2)=number of teeth for the first and second sun gears 52, 54, respectively; and

N_(P1), N_(P2)=number of teeth for the first and second planet gears 58, 60, respectively. For the embodiment of FIG. 6, N_(P1), N_(P2)=number of teeth for the first and second planet gear ends 72, 74, respectively.

φ=angular speed ratio of the crankshaft 12 to the input shaft 36, and is related to the creep rate though the following equation,

$\begin{matrix} {\phi = {\frac{2\left( {1 - \gamma_{0}} \right)}{1 - \gamma} \cdot}} & (5) \end{matrix}$

Substituting equation (5) into equation (3) and taking the derivative of (ω_(C)/ω_(S1)) with respect to γyields,

$\begin{matrix} {\frac{\partial\left( {\omega_{C}/\omega_{S\; 1}} \right)}{\partial\gamma} = {\frac{\left( {1 - \gamma_{0}} \right)\left( {1 - i_{b}} \right)}{\left\lbrack {\left( {1 - \gamma_{0}} \right) - {\left( {1 - \gamma} \right)i_{b}}} \right\rbrack^{2}} \cdot}} & (6) \end{matrix}$

The sensitivity of the speed ratio (ω_(C)/ω_(S1)) to creep rate at its nominal value γ₀ is

$\begin{matrix} {{\frac{\partial\left( {\omega_{C}/\omega_{S\; 1}} \right)}{\partial\gamma}_{\gamma = \gamma_{0}}} = \frac{1}{\left( {1 - \gamma_{0}} \right)\left( {1 - i_{b}} \right)}} & (7) \end{matrix}$

For i_(b)=0.96, γ₀=1%,

$\frac{\partial\left( {\omega_{C}/\omega_{S\; 1}} \right)}{\partial\gamma}_{\gamma = \gamma_{0}}{\approx {25\%}}$

Since variation in γ is generally dominated by low frequency components, compensation of any speed variation of the output shaft 42 caused by creep of belt 40 is possible by controlling the carrier 56. Several control structures are possible for achieving the desired angular position of the output shaft 42 device with respect to the position of the crankshaft 12. For example, the speed of the carrier 56 can be used as a control variable for a closed speed control loop to maintain the speed relationship set forth by equation (3). A controller operatively connects to the engine control unit ECU and the motor 50. The controller is configured to receive engine operating signals generated by the engine control unit ECU and to receive signals from position sensors 51 coupled to the input shaft 36 and to the output shaft 42 and in response thereto generates and sends a command signal to the motor 50 to command the motor 50 to control the planetary gear train 48 through the carrier 56 to adjust the phase shift angle between the camshaft 12 and the crankshaft 44.

FIG. 4 shows another control structure for achieving the desired angular position of the output shaft 42. In particular, FIG. 4 shows a torque-based control structure, generally shown as 92, suitable for use with the camshaft phasing device 46 of the present disclosure for achieving the desired angular position of the output shaft 42 with respect to the position of the crankshaft 12. The main control variable is the camshaft angle which is defined as the angular position of the camshaft 44 with respect to the position of the crankshaft 12. The control torque based control structure 92 comprises a controller 94 operatively connected to the engine E and the engine control unit 96 (ECU).

Based on information the controller 94 receives from the engine control unit 96, the controller 94 generates a torque command signal 98, such as a voltage signal. The received information includes, but is not limited to: a camshaft phase shift set point (reference); the actual camshaft phase shift angle measured from angular position sensor signals; a camshaft torque load and a camshaft angular position.

During operation, the actual camshaft phase shift angle is compared to a reference value to generate a differential (error) signal. The differential or error signal is then fed to a proportional-integral-derivative (PID) compensator 100 of the controller 94 to generate a feed back torque signal 102. This feed back torque signal 102, in turn, can be used to generate the torque command signal 98 to command the motor 50 to control to adjust the camshaft phase angle such that the error signal to the input of the PID compensator 100 or lead/lag compensator is reduced to an acceptable level. In doing so, the desired cam phase shift is achieved. For the torque-based control structure 92, the compensator 100 may comprise a proportional-and-derivative compensator (PD), a lead/lag compensator or a lead compensator.

During operation of the engine E, the control system may experience disturbances as the camshaft torque varies as a function of the cam phase angle during valve lift events. To improve the system's response to the reference input and increase robustness against disturbances, it is desirable to use a feed forward scheme to compensate for any known disturbances. Therefore, the controller 94 may further include a feed forward branch or block 104 for processing and computing the anticipated torque disturbances. The resulting feed forward torque signal 106 generated from the anticipated torque disturbance is fed forward to, and combined with, the output signal of the PID compensator 100 (or lead/lag compensator), forming the torque command signal 98.

The anticipated torque disturbance, also referred to as feed forward torque, is determined from two components, T_(rq) _(—) _(static) and T_(rq) _(—) _(friction). T_(rq) _(—) _(stalic) is calculated from the frictionless static equilibrium condition of the three-branch gear drive. T_(rq) _(—) _(friction) is the component required to overcome the frictional torque for current camshaft torque load. The sign of T_(rq) _(—) _(friction) is determined by the relative speed between the carrier 56 and the input shaft 36 (or the output shaft 42). For the disclosed configuration of the camshaft phasing device A, the feed forward torque is calculated as

T _(ffwd) =T _(rq) _(—) _(static) +T _(rq) _(—) _(friction)=(1−i _(b))·T _(com)+sgn(v)·f(T _(cam))   (8)

where T_(cam) is the camshaft torque load, which is a function of the phase angle of the camshaft.

The cam phase angle can be expressed by an analytical equation or as a look-up table. The function sgn(v) represents the sign of the relative speed v between the carrier 56 and the input shaft 36. The function f(T_(cam)) represents the magnitude of frictional torque T_(rq) _(—) _(friction). During operation, T_(ffwd) can be determined in dynamometer test as a function of engine torque and speed. The calibrated test data can then be stored in on-board memory devices (not shown) for real-time access.

During normal operation (a non-phase shifting event), the control structure 92 automatically controls the motor speed ω_(C) such that the speed relationship set forth by equation (3) is maintained. During a cam shift phase shifting event, the controller 94 adjusts the motor speed ω_(C) to cause the cam phase angle change over a small period of time to achieve the desired cam phase angle at the end of the shifting event.

As various changes could be made in the above constructions without departing from the scope of the disclosure, it is intended that all matter contained in the above description or shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense. 

1. In an internal combustion engine having an engine control unit, a crankshaft and a camshaft and a phase shift device coupling the crankshaft and the camshaft for controlling a phase shift angle between the crankshaft and the camshaft, the phase shift device comprising: a non-synchronous belt operatively connected to the crankshaft; an input shaft operatively connected to the non-synchronous belt, the input shaft having a first sun gear coupled to an end of the input shaft; an output shaft coupled to the camshaft, the output shaft having a second sun gear coupled to an end of the output shaft; a planetary gear train co-axially aligned around the first sun gear and the second sun gear, the planetary gear train includes a carrier and a planet gear having a first gear end and a second gear end that engage the first sun gear and the second sun gear, respectively, and are united to rotate about a common axis through a first bearing and a second bearing of the carrier; a motor operatively connected to the carrier; and a controller operatively connected to the engine control unit and the motor, the controller being configured to receive engine operating signals generated by the engine control unit and to receive signals from position sensors coupled to the input shaft and to the output shaft and in response thereto being configured to generate and send a command signal to the motor to command the motor to control the planetary gear train through the carrier to adjust the phase shift angle between the camshaft and the crankshaft.
 2. The phase device of claim 1 further comprising an input shaft pulley connected to the input shaft at an end of the input shaft opposite the first sun gear and comprises a crankshaft pulley connected to the crankshaft wherein the non-synchronous belt operatively connects to the crankshaft pulley and the input shaft pulley.
 3. The phase device of claim 2 wherein a creep rate of the non-synchronous belt is denoted “γ”; a ratio of pitch diameter of the crankshaft pulley to pitch diameter of the input shaft pulley is denoted “ψ”; and a ratio of the angular velocity of the crankshaft to the angular velocity of the input shaft is denoted “φ” which is characterized by the equation $\phi = {\frac{\psi}{1 - \gamma} \cdot}$
 4. The phase device of claim 3 wherein a gear ratio denoted “i_(b)” of the planetary gear train is characterized by the equation $i_{b} = \frac{N_{S\; 1} \cdot N_{P\; 2}}{N_{S\; 2} \cdot N_{P\; 1}}$ where N_(S1), N_(S2)=number of teeth for the first and second sun gears respectively; and N_(P1), N_(P2)=number of teeth for the first and second gear ends respectively.
 5. The phase device of claim 4 wherein the controller commands the motor to control the planetary gear train such that the angular speed of the carrier ω_(C) is controlled to maintain a relationship with the angular speed of the input shaft w_(S1) according to the equation: ${\frac{\omega_{C}}{\omega_{S\; 1}} = \frac{2 - {2i_{b}}}{\phi - {2i_{b}}}},$
 6. The phase device of claim 4 wherein the controller commands the motor to control the planetary gear train such that the angular speed of the carrier ω_(C) is controlled to maintain a relationship with the angular speed of the output shaft ω_(S2) according to the equation: $\frac{\omega_{C}}{\omega_{S\; 2}} = {\frac{\phi - {2i_{b}}}{\phi - {\phi \cdot i_{b}}} \cdot}$
 7. The phase device of claim 3 wherein the planetary gear comprises a first planet gear and a second planet gear such that the first planet gear meshes with the first sun gear and the second planet gear meshes with the second sun gear.
 8. The phase device of claim 7 wherein a gear ratio denoted “i_(b)” of the planetary gear train is characterized by the equation $i_{b} = \frac{N_{S\; 1} \cdot N_{P\; 2}}{N_{S\; 2} \cdot N_{P\; 1}}$ where N_(S1), N_(S2)=number of teeth for the first and second sun gears respectively; and N_(P1), N_(P2)=number of teeth for the first and second planet gears respectively.
 9. The phase device of claim 8 wherein the controller commands the motor to control the planetary gear train such that the angular speed of the carrier ω_(C) is controlled to maintain a relationship with the angular speed of the input shaft Ω_(S1) according to the equation: $\frac{\omega_{C}}{\omega_{S\; 1}} = \frac{2 - {2i_{b}}}{\phi - {2i_{b}}}$
 10. The phase device of claim 2 wherein angular speed ratio of the crankshaft to the input shaft is denoted “ω” and is related to the creep rate though the equation $\phi = \frac{2\left( {1 - \gamma_{0}} \right)}{1 - \gamma}$ where γ=the effective creep rate, defined as a percentage pitch line velocity loss with respect to pitch line velocity of the crankshaft pulley; and γ₀=a predetermined nominal creep rate of the non-synchronous belt.
 11. The phasing device of claim 1 wherein the controller comprises a feed forward block that is configured to process anticipated torque disturbances applied to the internal combustion engine.
 12. The phase device of claim 11 wherein an output of the feed forward branch T_(ffwd) is determined according to the equation T _(ffwd) =T _(rq) _(—) _(static) +T _(rq) _(—) _(friciton)=(1−i _(b))·T _(cam)+sgn(v)·f(T _(cam)) where T_(rq) _(—) _(static) is calculated from a frictionless static equilibrium condition of the three-branch gear drive; T_(rq) _(—) _(friction) is a component required to overcome frictional torque for current camshaft torque load; T_(cam)=the camshaft torque load; and f(T_(cam))=magnitude of T_(rq) _(—) _(friction).
 13. In an internal combustion engine, a method of controlling a phase shift angle between a camshaft and a crankshaft, the method comprising: connecting a non-synchronous belt to the crankshaft and to an input shaft having a first sun gear end coupled to an end of the input shaft; aligning a planetary gear train around the input shaft and around an output shaft coupled to the camshaft, the output shaft having a second sun gear coupled to an end of the output shaft; meshing a first planet gear of the planetary gear train with the first sun gear and meshing a second planet gear of the planetary gear train with the second sun gear, the first and second planet gears being united to rotate about a common axis through a carrier of the planetary gear train; operatively connecting a motor to the carrier; and commanding the motor to control the planetary gear train through the carrier to adjust the phase shift angle between the camshaft and the crankshaft.
 14. The method of claim 13 wherein controlling the motor comprises commanding the motor to control the planetary gear train such that the angular speed of the carrier ωw_(C) is controlled to maintain a relationship with the angular speed of the input shaft ω_(S1) according to the equation: $\frac{\omega_{C}}{\omega_{S\; 1}} = \frac{2 - {2i_{b}}}{\phi - {2i_{b}}}$ where a creep rate of the non-synchronous belt is denoted “γ”; a ratio of pitch diameter of the crankshaft pulley to pitch diameter of the input shaft pulley is denoted “ψ”; and a ratio of the angular velocity of the crankshaft to the angular velocity of the input shaft is denoted “co” which is characterized by the equation $\phi = \frac{\psi}{1 - \gamma}$ and where a gear ratio denoted “i_(b)” of the planetary gear train is characterized by the equation $i_{b} = \frac{N_{S\; 1} \cdot N_{P\; 2}}{N_{S\; 2} \cdot N_{P\; 1}}$ where N_(S1), N_(S2)=number of teeth for the first and second sun gears respectively; and N_(P1), N_(P2)=number of teeth for the first and second gear ends respectively.
 15. The method of claim 13 wherein controlling the motor comprises commanding the motor to control the planetary gear train such that the angular speed of the carrier ω_(C) is controlled to maintain a relationship with the angular speed of the output shaft ω_(S2) according to the equation: $\frac{\omega_{C}}{\omega_{S\; 2}} = \frac{\phi - {2i_{b}}}{\phi - {\phi \cdot i_{b}}}$ where a creep rate of the non-synchronous belt is denoted “γ”; a ratio of pitch diameter of the crankshaft pulley to pitch diameter of the input shaft pulley is denoted “ψ”; and a ratio of the angular velocity of the crankshaft to the angular velocity of the input shaft is denoted “φ” which is characterized by the equation $\phi = \frac{\psi}{1 - \gamma}$ and where a gear ratio denoted “i_(b)” of the planetary gear train is characterized by the equation $i_{b} = \frac{N_{S\; 1} \cdot N_{P\; 2}}{N_{S\; 2} \cdot N_{P\; 1}}$ where N_(S1), N_(S2)=number of teeth for the first and second sun gears respectively; and N_(P1), N_(P2)=number of teeth for the first and second gear ends respectively.
 16. In an internal combustion engine, a method of controlling a phase shift angle between a camshaft and a crankshaft of an internal combustion engine, the method comprising: connecting a non-synchronous belt to the crankshaft and to an input shaft having a first sun gear end coupled to an end of the input shaft; aligning a planetary gear train around the input shaft and around an output shaft coupled to the camshaft, the output shaft having a second sun gear coupled to an end of the output shaft; meshing a first planet gear of the planetary gear train with the first sun gear and meshing a second planet gear of the planetary gear train with the second sun gear, the first and second planet gears being united to rotate about a common axis through a carrier of the planetary gear train; operatively connecting a motor to the carrier; receiving an angular position signal of the camshaft; comparing the camshaft pahse signal signal to a reference signal provided by an engine control unit; and generating a torque command signal based on the compared camshaft signal wherein the torque command signal commands the motor to adjust the phase shift angle between the camshaft and the crankshaft.
 17. The method of claim 16 wherein the torque command signal is denoted “T_(ffwd)” and is determined according to the equation T _(ffwd) =T _(rq) _(—) _(static) +T _(rq) _(—) _(friciton)=(1−i _(b))·T _(cam)+sgn(v)·f(T _(cam)) where T_(rq) _(—) _(static) is calculated from a frictionless static equilibrium condition of the three-branch gear drive; T_(rq) _(—) _(friction) is a component required to overcome frictional torque for current camshaft torque load; T_(cam)=the camshaft torque load; and f(T_(cam))=magnitude of T_(rq) _(—) _(friction). 